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Nonlinear Combustion Instability in

Liquid-Propellant Rocket Motors

Samuel Z. Burstein and Wallace Chinitz

Second Quarterly Report

to . -

Jet Propulsion Laboratory

January 30, 1968

JPL COMTBACT 951946

"This work was performed for the Jet Propulsion Laboratory, California Institute of Technology, as sponsored by the National Aeronautics and Space Administration under Contract NAS7-100."

https://ntrs.nasa.gov/search.jsp?R=19680009979 2018-08-29T01:57:03+00:00Z

"This report contains information prepared

by Mathematical Applications Group, Inc.

(MAGI) under JPL subcontract. Its content

is not necessarily endorsed by the Jet

Propulsion Laboratory, California Institute

of Technology, or the National Aeronautics

and Space Administration."

i

TABLE OF CONTENTS

INTRODUCTION ...................................... 1

DROPLET EVAPORATION AND COMBUSTION ANALYSES ....... 2 A . The Rate of the N2H4/N204 Reaction ........... 2 B . The Diffusion-Controlled Flame Analysis ..... 11

. FLUID DYNAMIC MODEL .............................. 17 A . First-Row Mesh Point Calculation ............ 17 B . An Exact Solution-Comparison with

Machine Experiment ......................... 19 C . A Finite Amplitude Calculation of a

Rotating Wave .............................. 21 D . Convergence of the Difference Equations ..... 2 3 E . Simple Forcing Function ..................... 2 6 F . Program COMB Status ......................... 28

ii

INTRODUCTION

During the second quarter of the contract there has been

significant progress in the construction of a mathematical model

describing the dynamics of a rotating finite-amplitude wave in a

cylindrical combustion chamber. This quarterly report describes

some of the details of this activity.

The first section describes the theory of finite rate proc-

esses in the combustion of hydrazine (NzHq)/nitrogen tetroxide

(N2O4);

for the modified surface flame model of combustion. Also discussed

the results will be used to predict kinetic parameters

in this section are some results obtained from the diffusion flame

analysis.

The second section is involved with the fluid dynamic model.

A new treatment for the calculation of special mesh points in

COMB is presented. Most important, however, the results of some

time dependent computations are described in which a finite am-

plitude pressure wave is allowed to rotate and steepen in a

cylindrical chamber. The reader should refer to reference 1 for

background material and additional references (especially reference

2 0 ) .

The authors would like to acknowledge the significant contri-

bution of Harold Schechter in the organization, analysis and

programming of the digital programs being developed in this

research program.

1

I. THE DROPLET EVAPORATION AND COMBUSTION ANALYSES

A . The Rate of the N2H4/N204 Reaction

It will be recalled (Ref. 1) that the species conservation

equation including finite rate chemical reaction can be written:

dYi - d 2 dYi 2 N F yNo (1) a- ( n -) = - iin bYF dn 0

where a = the dimensionless mass burning rate

Yi = mass fraction of species i

n = radius/droplet radius

Mi = molecular weight of species i

p = mass density

NF,N, = reaction orders 2

b = rDk/D

k = Arrhenius reaction rate constant = A(T)exp(-E/RT)

D = diffusion coefficient

It was also shown (Ref. 1) that by neglecting the energy

transport due to concentration gradients (the Dufour effect) and

assuming the Lewis number to be unity, the energy equation takes

on the same form as equation (1) above:

2 NF NO a- - - - i q b Y F (,,* dt -

dn F

dt dn dn -1 -

2

In order to apply Peskin's (Ref. 2) modified flame surface

analysis, realistic estimates must be obtained for the reaction

rate constant, k = A (T) exp (-E/RT) , and the reaction orders, NF and No, corresponding to the N2H*/N204 reaction. These can

be obtained from a one-dimensional, finite-rate analysis of the

oxidation process, using the simplified chemical kinetic mechanism

deduced from Sawyer, and discussed in Ref. 1. This method is

formulated in the following manner.

As mentioned above, the flow equations are written for one-

dimensional, constant pressure, inviscid flow along streamtubes,

with the

a.

b.

C.

following additional assumptions:

Negligible diffusion normal to the flow direction,

leading to a uniform mixture of (gaseous) species

at any streamwise point. Therefore, the oxidation

is taken to be 'reaction-controlled'.

The flow is steady and adiabatic.

The initial concentrations of the oxidizing species

are obtained in accordance with the following

reaction equation and with the assumption of

chemical equilibrium:

N2O4 + a N204+bN02 + cNO + d02 ( 3 ) The method for obtaining the equilibrium species concentrations

is detailed in Appendix A.

With these assumptions, the relevant equations are:

Energy Conservation

dh = ; h = constant dt

(4)

where h includes sensible - and chemical enthalpy.

3

Species Conservation

dYi = ri Pdt (5)

Equation of State

P = p M RT

Auxiliary Equations

h = C Yihi i

M = C Yi (i F)

The enthalpy terms are taken to be linear in form: -

hi = Ai + c Pi

where the reference temperature was chosen to be 1000'K. Values

for Ai and E

chemical kinetic mechanism and t h e species generation terms, ri,

are in Tables I and 11, respectively. The reaction rate constants

are in Table I.

were obtained from the data of Reference 5. The Pi

It should be emphasized that this premixed, homogeneous gas-

phase analysis is designed to supply an appropriate representation

of the overall (global) oxidation reaction rate, for use in equa-

tions (1) and ( 2 ) . It is not, of course, a representation of the

processes occurring around the fuel drop, where a counter-flow

diffusion flame exists, as described by equations (1) and ( 2 ) .

Results for two values of the ambient pressure are shown in

Figures 1 and 2 . At one atmosphere (Figure 11, and an initial

temperature of 1500'K, the reaction is seen to commence at about

seconds, continue smoothly until about seconds after

4

b

. which a rapid rise in temperature is noted. If the calculations

were continued, a rapid leveling off to the adiabatic flame tem-

perature would be noted. In this case, the ignition delay time

can be estimated to be seconds.

At the lower initial temperature (Ti = 1000K), the ignition

delay time is seen to increase in Figure 1 to about S X ~ O - ~ seconds.

When the pressure is increased to 20 atmospheres (Figure 2 ) ,

a somewhat different behavior is noted. At both initial temper-

atures, the reaction is seen to commence quite early (at about

10 seconds for Ti = 1500K, and before lo seconds for Ti =

1000K). However, a pause in the reaction then occurs and, for

the case of Ti = 1500K, between 5x10- and 5x10-* seconds little

temperature change is noted. After 5x10-* seconds, a smooth and,

ultimately, rapid rise in temperature is noted.

The reason for this behavior can be explained by reference

to Figure 3 and Table 11. In Figure 3, it is seen that NO2 dis-

appears rapidly during this initial temperature rise, with a

corresponding decrease in the concentration of hydrazine. This

same behavior was not noted f o r the case of p = 1 atm because

reaction number 2 in Table I1 is second-order and, hence, pre-

dominates at higher pressures. An accompanying increase of NO,

as a result of reaction 2, can be seen in Figure 3 as well.

After the depletion of the N02, the N2H4/NO and N2H4/O2

oxidation reactions take over and proceed as first-order reactions

(Table 11), leading to a second delay period, and a subsequent

rapid reaction at about seconds. The behavior of NH3 and

6

H 2 is shown in Figure 4 . It can be seen that they rise in concen-

tration initially, as a consequence of the hydrazine decomposition

reaction (reaction 1, Table 11), then rapidly disappear when the

main reaction occurs.

It is clear then, that the 'ignition delay time' is difficult

to define in the pressure regime where this two-step reaction

occurs. It can be anticipated that the traditional plot of igni-

tion delay as a function of reciprocal temperature will not produce

a straight line since the rate-controlling reactions are different

at different pressures. This will lead to the requirement for a

different 'global' reaction rate constant for the high and low

pressure regimes. Current efforts are directed toward obtaining

sufficient computer runs so that the chemical kinetic parameters

can be determined in both pressure regimes.

9

0 l-i

U I

B. The Diffusion-Controlled Flame Analysis

The diffusion-controlled flame analysis which was described

in Reference 1 has been coupled to the basic computer program

COMB. In this section, we compare the results obtained from t

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